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Topic: 2+2=4 ... How?
Replies: 9   Last Post: Sep 29, 2012 10:40 AM

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: 2+2=4 ... How?
Posted: Sep 29, 2012 10:40 AM
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Barb Knox wrote:
>
> In article <op.wk94sq2i1hq4pq@leon-hp>, wilson <winslole@udayton.edu>
> wrote:
>

> > On Thu, 27 Sep 2012 01:40:34 -0400, wilson <winslole@udayton.edu> wrote:
> >

> > > On Thu, 27 Sep 2012 01:18:28 -0400, Brian M. Scott <b.scott@csuohio.edu>
> > > wrote:
> > >

> > >> On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur
> > >> <madhur.varshney@gmail.com> wrote in
> > >> <news:dca7ee3d-9446-4fcc-b4fb-c01fdbc685ea@googlegroups.com>
> > >> in alt.math.undergrad:
> > >>

> > >>> The natural numbers that we use are said to be derived
> > >>> from what so called Peano's Axioms.

> > >>
> > >> They *can* be; this is not the only possible formal
> > >> foundation for them.
> > >>

> > >>> While these axioms (listed below) give a method of
> > >>> building up counting numbers they do not define or
> > >>> construct basic arithmetic operations like addition,
> > >>> subtraction, multiplication, etc or basic comparisons
> > >>> like that of equality.

> > >>
> > >> Equality is assumed to be a known relation. The arithmetic
> > >> operations and the linear ordering on the natural numbers
> > >> are defined using the axioms. This is explained, albeit
> > >> briefly, in the Wikipedia article on the Peano axioms:
> > >>
> > >> <http://en.wikipedia.org/wiki/Peano_axioms?banner=none#Arithmetic>
> > >>
> > >> [...]
> > >>
> > >> Brian

> > >
> > > A bit more complicated:
> > >
> > > First you need another axiom. From the Wikipedia article:
> > >
> > > Addition is the function + : N Å~ N Å® N (written in the usual infix
> > > notation, mapping elements of N to other elements of N), defined
> > > recursively as:
> > >
> > > a + S(0) = a
> > > a + S(b) = S(a+b)
> > > Now can define 1 as S(0), 2 as SS()) and 4 as SSSS(0).
> > >
> > > the proof that 2 + 2 = 4 is then a matter of substituting the right
> > > thing in the right place.

> >
> >
> > sorry. my mistake. Define 2 as SS((0)).

>
> Also, a + 0 = a, instead of a + S(0) = a.


Yebbut note that Madhur's post made no reference 0, for her the natural
numbers begin at 1.

--
Where are the songs of Summer?--With the sun,
Oping the dusky eyelids of the south,
Till shade and silence waken up as one,
And morning sings with a warm odorous mouth.



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